Question: What integer value of $n$ will satisfy $n + 10 > 11$ and $-4n > -12$?
Answer: We treat these inequalities one-at-a-time first.  Subtracting 10 from both sides of the first inequality simplifies it to \[n>1.\] To deal with the second inequality, we divide both sides by $-4$, being sure to reverse the inequality sign: \[n<3.\]

Luckily, there is only one integer that solves both of these inequalities, namely $\boxed{2}$.